mode. The system can also be used in the sequential mode if desired. The principal advantage of the sequential mode of operation is that it allows the operator to change parameters between determinations to ensure that optimum conditions for each element are obtained. The multiplexing mode, on the other hand, has the advantage of a reduced analysis time which allows multielement analysis on a transient atomic vapor plume. However, a compromise has to be made as to the parameters chosen. The TDM method gave detection limits comparable to the sequential, dispersive method only for separated air-HZ and Ar-HZ flames. For other flames, the nondispersive method is inferior because the increase in energy throughput is accompanied by a greater increase in noise throughput.
T o take full advantage of the TDM system, a low background, yet high atomization efficiency atomizer should be employed. A good combination would seem to be a TDMAF system with a graphite filament nonflame atomizer. Work in this area is now under way in these laboratories.
ACKNOWLEDGMENT The authors thank Charles Hacker of the Michigan State University Department of Chemistry machine shop for construction of the lamp holder and burner.
RECEIVEDfor review February 20, 1974. Accepted August 12, 1974.
Precision of Atomic Absorption Spectrometric Measurements J. D. ingle, Jr. Department of Chemistry, Oregon State University, Corvallis, Ore. 9733 1
A detailed theoretical study of the factors which affect the relative precision of atomic absorption (AA) measurements is presented. The theory takes into account shot and flicker noise in various signal and background sources of radiation, readout noise, flame transmission noise, and sampling or atomization reproducibility. In many cases, the general equations can be simplified to limiting expressions if one type of imprecision is dominant. It is shown that the optimum range to make AA measurements depends on what factors limit the precision. Application of the theory to typical AA inslrument systems and also photon counting, non-flame, dualwavelength, and vidicon AA systems is also discussed.
The sources of noise in atomic absorption measurements have been discussed by a number of authors (1-6). Since noise often limits measurement precision, knowledge of the origins of the major sources of noise provides information that can be used for optimization of experimental variables. Knowledge of how the relative standard deviation in the absorbance (u*/A) or the signal-to-noise ratio (S/N) varies with analyte concentration ( c ) is important if standard and unknown solutions are to be adjusted to optimal concentration ranges. It is unfortunate that, of the great majority of abundant papers which deal with atomic absorption spectrometric analysis, only a small fraction present or discuss precision ( i e . , report standard deviations). If precision data are presented, they are often with reference to the detection limit rather than at concentrations or absorbances at which analyses are usually run. Finally, even if substantial precision data are presented, little attempt is made to identify the causes of the imprecision such as the type of noise that limits precision. Such information, if measurements were (1)J. D. Winefordner and T. J. Vickers. Anal. Chem., 36, 1947 (1964). (2)J. D. Winefordner and T. J. Vickers, Anal. Chem., 36, 1939 (1964). (3)M. L. Parsons, W. J. McCarthy, and J. D.Winefordner, J. Chem. Educ., 44, 214 (1967). (4)J D. Winefordner, V. Svoboda, and L. J. Cline, CRC Crit. Rev. Anal. Chem., August 1970. (5) W. Lang and R . Herrman, Optik, 20, 347 (1963). (6)J. D. Winefordner and C. Veillon. Anal. Chem. 37,416 (1965).
made under optimal conditions, would be useful to compare analyses of a given sample by two researchers or on two instruments. A number of workers (1-6) have dealt extensively with the types of noise and S/N near the detection limit and the dependence of the noise sources on various instrumental and chemical parameters. Such work has proved extremely valuable for comparison of AA to other flame techniques and for optimization of measurement conditions a t the limit of detection. The factors which influence the precision at concentrations significantly greater than the detection limit can be different from those factors a t the detection limit. Hence, optimization of experimental variables at the detection limit may not provide optimal conditions a t higher concentrations. The little work (7-11) which has been done to predict how U A / Avaries with A for AA measurements has referred for the most part to the older theories developed for molecular absorption measurements in which only reading errors or errors independent of T are considered, so that a minimum for o*/A is predicted at about 37% T . More recent discussions (12, J3) of molecular absorption have shown that the u*/A us. A curve predicted by the assumption of a constant error in T can be greatly in error if direct absorbance readout is used or if readout resolution is sufficient and noise dependent on T i s significant. Published u*/A us. A data (9, 14-18) for AA indicate that both the magnitude of UAIAand the shape of the UAIA curve depend on the element analyzed, even with the same instrument, and on the instrumental variables chosen for a (7)Walter Slavin, "Atomic Absorption Spectroscopy," Wiley-lnterscience. New York, N.Y., 19_68,pp 66-68. (8)Juan Ramierz-Munoz, "Atomic Absorption Spectroscopy," Elsevier Publishing Co., New York, N.Y., 1968,pp 167-259. (9)H. Khalifia, G.Syehla, and L. Erdey, Talanta, 12, (1965). (IO)J. Ramierez-Munoz, Microchem J., 12, 196 (1967). (11) I. Rubeska and B. Moldan, "Atomic Absorption Spectrophotometry," CRC Press, Cleveland, Ohio, 1969,pp 84-85. (12)J. D.Ingle. Jr., and S. R. Crouch, Anal. Chem., 44, 1375 (1972). (13)J. D.Ingle, Jr., Anal. Chem., 45, 861 (1973). (14)D. R . Weir and R . P. Kofluk, At. Absorption Newslett., 6, 24 (1967). (15)B. Meddings and H. Kaiser, At. Absorption Newslett., 6, 28 (1967). (16)J. T. H. Roos. Spectrochim. Acta, 248, 255 (1969). (17)J. T. H. Roos, Spectrochim. Acta, 258, 539 (1970). (18)J. T.H. Roos, Spectrochim. Acta, 288, 407 (1973).
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 1 4 , DECEMBER 1974
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Table I. Signal Voltages and Currents E , = i,mG = analyte transmission voltage due to r a diation from source passed b y the flame with the analyte solution aspirating into the flame, V wz = current gain of the photomultiplier, dimensionless is = sample signal photocurrent due to transmission of source radiation by flame with the analyte solution aspirating into the flame, A G = amplification factor for amplifier-readout system, V-A-I. G takes into account the response of the amplifier readout system to the r m s photoanodic signal and is frequency dependent. G i s 0 for dc signals in a modulated system. E , = i,mG = reference transmission voltage due to radiation from source passed by the flame with the blank solution aspirating into the flame, V i, = reference signal photocurrent due to transmission of source radiation by flame with the blank solution aspirating into the flame, A E,, = ib,mG =background emission voltage due to flame and concomitant background radiation from thermal emission of non-analyte species, V i, = background emission photocathodic current, A E , = i,mG = dark current voltage, V id = effective cathodic dark current, A E , = i,mG = analyte emission voltage due to radiation from the thermal emission of the analyte species, V i, = analyte emission photocathodic current, A given element. Such evidence suggests that the simple theory may not always be adequate. Recently (19, 20) photon counting was used for atomic absorption measurements. Equations that essentially estimate aAIA based on signal quantum noise (19) were used. The comparison of theoretical and experimental relative standard deviations indicated that other noise sources in addition to quantum noise must be considered. Roos (16-18) and Price (21) have presented theories which deal with the precision of AA measurements and have identified four limiting cases in which the random error of a transmittance measurement ( A T ) is independent of T , proportional to T , proportional to v'T, and proportional to TA. The reasons for these dependencies were discussed and the theoretical equations were fitted empirically to experimental data (16-18). Experimental data often were found to fit the fourth limiting case which indicated that variations in the absorption characteristics of the analyte in the flame may be limiting. In this paper equations are developed which predict the dependence of oAIA on A or T. These equations take into account all of the major sources of noise that might be significant in AA measurements as well as reading error. The theoretical expressions are formulated in terms of various (19) R. M. Daynall, 9. L. Sharp, and T . S. West, Talanta, 19, 1942 (1972). (20) M. K. Murphy, S. A. Clyburn, and C. Veillon, Anal. Chem., 45, 1468 (1973). (2 1) W. J. Price, "Analytical Atomic Absorption Spectrometry," Heyden and Son Ltd.. Londo'n, 1972, pp 103-1 1 1 .
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measured or calculated experimental parameters so that they can be readily used. In many cases, the equations can be considerably simplified if shot or flicker noise from a particular component of the measured signals is limiting. Factors which cause differences among solutions, such as random errors in sample preparation (e.g., pipetting error) or contamination, are not considered. It has been assumed that background fluorescence is negligible. The system to be described is a modern single beam AA spectrometer with a modulated line source (typically a hollow cathode tube), a nebulizer-burner, a monochromator, a photomultiplier with a stabilized power supply, suitable signal modifier electronics for conversion, amplification, and filtering of the modulated photomultiplier signal, and a voltage readout device. The equations presented for this system are simplifications of those equations derived for very general situations and are presented in the appendix.
SIGNAL EXPRESSIONS Transmittance Readout. For transmittance readout instruments, three voltage measurements are needed to obtain the transmittance: E,, = the 0% T voltage or the total output voltage with the light source off and the blank aspirating into the flame, V E,, = the 100% T voltage, the reference voltage, or the total output voltage with the light source on and the blank aspirating into the flame, V E,, = the sample voltage or the total output voltage with the light source on and an analyte solution aspirating into the flame, V The transmittance is given by
The three measured signal voltages can be broken down into their component voltages as
E,, = E ,
-I-E,
E,,
where all terms are defined in Table I. All currents defined in Table I are effective root mean square (rms) cathodic currents. The effective photocathodic current is the current from the photocathode of the photomultiplier that reaches the first dynode and causes secondary emission and, hence, takes into account the collection efficiency of the first dynode. The dark current is assumed to originate solely from the photocathode due to thermal emission (22). A given component of the anodic current can be found by multiplication of the appropriate cathodic current by m. The analyte fluorescence signal has been assumed to be negligible compared to E,, although in the appendix expressions are developed for cases where the analyte fluorescence signal is not negligible as well as for an instrument with an unmodulated source and dc electronics. With modulation, the mean value of dc signals independent of the modulated light source radiance, such as analyte emission ( E e ) ,flame and concomitant background emission ( E be), and dark current voltage ( E d ) , can be set to zero in Equations 2-4, so that
T = E,/E, = is/& (22) J. D. Ingle, Jr., and S.R. Crouch, Anal. Chem., 44, 785 (1972).
A N A L Y T I C A L CHEMISTRY, V O L . 46, N O . 14, DECEMBER 1974
(5)
The amplifier does respond to noise in these dc voltages that occurs over the bandwidth of the amplifier centered a t the modulation frequency. It is common to use a suppression voltage to set Eot to the zero (0% T ) of the readout device and the gain (amplifier (C) or photomultiplier ( m ) )controls to set E, to full scale (100% T ) of the readout device. Under these conditions E , can be read directly as T . Absorbance Readout. Many modern systems provide readout directly in absorbance by use of logarithmic ratio circuitry. Under these conditions only two measurements are used to measure the absorbance:
E l , = -k' log(E,/k") (6) = s a m p l e a b s o r b a n c e voltage o r voltage with a n a l y t e solution a s p i r a t i n g into t h e f l a m e ,
V E,, = -k' log(E,/k")
(7)
z e r o absorbance voltage, reference a b s o r b a n c e v o l t a g e , o r voltage with t h e b l a n k s o l u t i o n a s p i r a t i n g into t h e f l a m e ,
V where k" is the constant reference voltage and k' is the logarithmic conversion constant in V/decade. Their difference is a voltage EA which is proportional to the analyte absorbance:
E , = E,, - E l , = -k' log(E,/E,)
= -k' log T (8)
Usually El, is set equal to the zero of the readout device with a suppression voltage and gain controls.
NOISE EXPRESSIONS The standard deviation in the measured absorbance (UA) is related to the standard deviation of the three voltage measurements (Ert,Est,Eat) needed to obtain T for transmittance readout instruments or to the two voltage measurements (El, and El,) necessary to obtain E A for absorbance readout instruments. The imprecision in these voltage measurements is due primarily to noise associated with each of these voltages. Each of the three measured voltages was broken up into its component voltages (Equations 2-4) in order to identify fully the origins of each voltage and to provide a means to keep track of sources of noise. Hence, one can assocrate one or more types of noise with each component voltage (E,, E,, E,, Et,,, Ed) of which Est,Ert,and E,, are constituted. The variance in each component voltage is due to shot noise and flicker noise. The shot noise can be broken down into quantum and secondary emission noise. The variance due to shot noise can be easily calculated and represents the fundamental limit in measurement precision. Flicker noise accounts for non-ideal or non-fundamental fluctuations (ie., above shot noise) in each of the above component voltages. Unlike shot noise, which is white, the flicker noise is often pink or l / f type noise. It is assumed that the flame transmission flicker factor &), the background emission signal ( i b e ) , and the background emission flicker factor (x)are independent of the analyte concentration and absorbance. If any of the above variables is absorbance dependent, the dependence can be incorporated into the developed expressions. It also is assumed that fluctuations in the gain of the photomultiplier ( m ) above secondary emission noise, fluctuations in the gain ( G ) of the amplifier system, and unidirectional drifts between sample, reference, and 0% T measurements are negligible. A variance (uar) also can be associated with the amplifier-readout system to take into ac-
count the noise present in a signal voltage in the absence of the shot and the flicker noises in the component voltages. The variances in Ert,Est, and Eot are shown as the sum of a number of variance terms in Equations 9-15 in Table I1 where it is assumed that all noise sources are statistically independent. The relationship of each of these component variances to instrumental parameters (22) is also listed in Table 11. (See equations 16-36.) Note that three independent flicker factors are used to describe non-fundamental fluctuations in E, and E,. This is necessary to identify clearly the different causes of flicker since in a given situation one type of flicker may predominate. The flicker factor El is the lamp flicker factor that accounts for the non-ideal fluctuations in the source radiance of the particular lamp used due to the lamp characteristics or vibrations and is independent of the flame. The flicker factor ( 2 takes into account non-ideal fluctuations in transmission characteristics of the flame and depends on the sample solution matrix and flame composition. The transmission of the flame is due both to absorption and to scattering by non-analyte species. The scattering is greater and hence & is more significant for lower wavelengths. Finally, .5*and E3 are the analyte absorption flicker factors which take into account non-ideal fluctuations in the absorption and atomization characteristics of the analyte which are caused by variations in the concentration of neutral ground state atoms viewed, in the absorptivity of the analyte, in the flame path length, or in scattering by the analyte. These fluctuations are due to the dynamic nature of the flame and to factors such as slow fluctuations in aspiration rate, in the gas flow rates, in the atomization efficiency, and in the flame size or position. This flicker factor is written in two forms since it is useful to know how sampling reproducibility affects both A and
E,. FINAL EXPRESSIONS FOR THE RELATIVE STANDARD DEVIATION OF THE ABSORBANCE Transmittance Readout Complete Expression. The standard deviation in the absorbance, UA,is related to the standard deviation in the measured transmittance, UT, and the standard deviation in transmittance is related to the standard deviations in the three signal voltages, ust, art, and uot, by Equations 37 and 38, respectively, (12, 23) if UT > (3 e. ( E l 2 (22)'/2 = C3 = 0.01 f (E12 t 2 2 ) " 2 = ( 3 = 0.001 g. (('2 t2')"2 = 0.001, ( 3 = 0.01 h. (ti2+ (22)1'2 = 0.01, ( 3 = 0.03
+ + + + +
uA/A = {(i, In T)-'[Ki,(l
+
A
A
Ti) + (ae,/mGT)2 +
This case represents the lowest possible U A / Afor a given reference photocathodic current (&). Note that (K/ir)l/zis just the relative standard deviation or relative rms noise in the reference signal. Curves a and b in Figure 2 are plots in which the relative rms noise in E , is 1.0 and 0.1%, respectively (0.0043 and 0.00043 absorbance unit, respectively). The signal photocurrents indicated are calculated for a 1-Hz bandwidth and represent typical to below average signals normally expected. Instrumentally, K , which is proportional to the noise equivalent bandwidth ( A f ) , is reduced by increasing the time constant or integration time of the amplifier-readout system. For larger signal photocathodic currents or smaller noise equivalent bandwidths, other sources of noise are exUA/A pected to be more significant (Le., for (K/ir)l/*I < 0.1% for A > 0.04 from Equation 48). Here the equations indicate how the reference current and noise bandwidth must be adjusted so that signal shot noise is not limiting. Case I1 may apply a t low absorbances, but a t higher absorbances, Case I variance terms, analyte emission noise, or analyte absorption flicker noise may be important. Note that the highest precision is achieved near an absorbance of one. Flicker Noise Limit. Under conditions of large signal photocathodic currents or small noise equivalent band5 widths, when signal shot noise is small ( i e . (~Y/i,)~l~ and analyte emission noise is negligible, source or transmission flicker noise or analyte absorption flicker may become dominant, and Equation 46 reduces to
Case 111-Source or Transmission Flicker Limited. Equation 49 reduces to Equation 50 if analyte absorption flicker is negligible and measurements are source or transmission flicker limited.
a,/A Equation 46 should apply to most AA instruments if readout resolution can be made negligible by use of high resolution digital voltmeters or scale expansion so that signal shot noise, source or flame transmission flicker noise, analyte emission noise, or analyte absorption flicker noise is obvious and limits the measurements. Under these conditions, the expressions are independent of the mode of readout. With our transmittance readout instrument, U A due to uot was found to be 2 X 10-5 absorbance unit or less and negligible compared to other sources of noise. The remainder of the limiting cases discussed below will be special cases of Equation 46. Detection Limit. Near the detection limit, T N 1, i,
